﻿ 输入饱和非线性切换Hamilton系统镇定与<i>H<sub>∞</sub></i>控制
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 山东大学学报(工学版)  2016, Vol. 46 Issue (3): 79-86  DOI: 10.6040/j.issn.1672-3961.0.2015.394 0

### 引用本文

LI Han, WEI Airong. Stabilization and H control for nonlinear switched Hamiltonian systems subject to actuator saturation[J]. Journal of Shandong University(Engineering Science), 2016, 46(3): 79-86. DOI: 10.6040/j.issn.1672-3961.0.2015.394.

### 文章历史

Stabilization and H control for nonlinear switched Hamiltonian systems subject to actuator saturation
LI Han, WEI Airong
School of Control Science and Engineering, Shandong University, Jinan 250061, Shandong, China
Abstract: The stabilization and H control problems of nonlinear switched Hamiltonian systems (NSHSs) subject to actuator saturation under arbitrary switching paths was investigated. The stabilization of NSHSs with actuator saturation was studied by designing suitable state feedback and deriving a sufficient condition. Then, an H controller was designed to realize the H control of NSHSs subject to actuator saturation with external disturbances, under the designed controller, γ-dissipation inequality held for the system with external disturbances, moreover, the globally asymptotical stabilization was achieved for the corresponding closed-loop system without external disturbances. A simulation example showed that the proposed method and results were effective.
Key words: Hamiltonian systems    actuator saturation    nonlinear switched    stabilization    H control
0 引言

1 预备知识

 $\dot{x}=[{{J}_{\lambda (t)}}\left( x \right)-{{R}_{\lambda (t)}}\left( \text{ }x \right)]\nabla {{H}_{\lambda \left( t \right)}}(x),$ (1)

2 输入饱和非线性切换 Hamilton 系统镇定

 $\dot{x}=[{{J}_{\lambda (t)}}\left( x \right)-{{R}_{\lambda (t)}}\left( x \right)]\nabla {{H}_{\lambda (t)}}\left( x \right)+{{g}_{\lambda (t)}}\left( x \right)sat\left( u \right),$ (2)

 \text{sat}\left( {{u}_{i}} \right)=\left\{ \begin{align} & {{\rho }_{i}},{{u}_{i}}>{{\rho }_{i}} \\ & {{u}_{i}},-{{\rho }_{i}}\le {{u}_{i}}\le {{\rho }_{i}},i=1,2,\cdots ,m \\ & -{{\rho }_{i}},{{u}_{i}}<-{{\rho }_{i}} \\ \end{align} \right.。 (3)

 ${{\eta }^{T}}\eta \le \varepsilon {{u}^{T}}u,$

 $\eta =\text{sat}\left( u \right)-u,$ (4)

 $u=u{{|}_{\lambda \left( t \right)=i}}=-g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right),i\in \Lambda ,$ (5)

 $\dot{x}=[{{J}_{i}}\left( x \right)-{{R}_{i}}\left( x \right)]\nabla {{H}_{i}}\left( x \right)+{{g}_{i}}\left( x \right)\text{sat}(-g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}(x))$

 \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)=\nabla H_{i}^{\text{T}}\left( x \right)\dot{x}=-\nabla H_{i}^{\text{T}}~\left( x \right){{R}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right)- \\ & \nabla H_{i}^{\text{T}}~\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta \\ \end{align}

 \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)\le -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)+{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}\left[ \nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+{{\eta }^{\text{T}}}\eta \right]\le \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)+{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}{{\varepsilon }_{i}}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)= \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)+\frac{1}{2}\left( 1-{{\varepsilon }_{i}} \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right) \\ \end{align}。

3 输入饱和非线性切换Hamilton系统H控制

 \left\{ \begin{align} & \dot{x}=[{{J}_{\lambda (t)}}\left( x \right)-{{R}_{\lambda (t)}}\left( x \right)]\nabla {{H}_{\lambda (t)}}\left( x \right)+{{g}_{\lambda (t)}}\left( x \right)\text{sat}\left( u \right)+{{{\bar{g}}}_{\lambda (t)}}\left( x \right)\omega , \\ & z={{r}_{\lambda (t)}}\left( x \right)g_{\lambda (t)}^{\text{T}}\left( x \right)\nabla {{H}_{\lambda (t)}}(x), \\ \end{align} \right. (6)

R1: 系统存在外部干扰(ω≠0)时,对每1个 iΛ,γ-耗散不等式${{\dot{H}}_{i}}\left( x \right)+{{Q}_{i}}\left( x \right)\le \frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right),\forall \omega$,沿由控制率u|λ(t) 和系统 (6) 所构成的闭环系统的所有轨线都成立,其中,Qi(x)≥0为一标量函数;

R2: 无外部干扰(ω=0)时,闭环系统渐近稳定。

(1) ${{T}_{i}}\left( x \right)\ge 0,\forall i\in \Lambda$,其中,${{T}_{i}}\left( x \right):{{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{\bar{g}}_{i}}\left( x \right)\bar{g}_{i}^{\text{T}}\left( x \right)+$$\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)-$$\frac{1}{8}{{\varepsilon }_{i}}{{g}_{i}}\left( x \right){{\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right)$;

(2) 存在矩阵Sl(x)>0,lΛ,而其他Si(x)≥0,∀ilΛ,其中,{{S}_{i}}\left( x \right):={{R}_{i}}\left( x \right)+\frac{1}{2}{{g}_{i}}\left( x \right){{\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)-\frac{1}{8}{{\varepsilon }_{i}}{{g}_{i}}\left( x \right){{\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right),则在任意切换路径 λ(t)下,系统 (6) 的 H控制器  u=u\left| _{\lambda \left( t \right)=i}=-\left[ \frac{1}{2}r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{I}_{m}} \right]g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right) \right.,i\in \Lambda , (7) 其中:Imm×m 阶单位阵,并且取Qi(x)=∀HiT(x)Ti(x)∀Hi(x)≥0,∀i∈Λ。 证明 设λ(t)=iΛ 为任意切换路径,系统存在外部干扰(ω≠0)时,由系统 (6) 得  \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)=-\nabla H_{i}^{\text{T}}\left( x \right)\dot{x}=\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{J}_{i}}\left( x \right)-{{R}_{i}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right) \\ & +\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\text{sat}\left( u \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\omega \\ \end{align}。 (8) 将式 (7) 代入式 (8),并由式 (4) 得  \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)=-\nabla H_{i}^{\text{T}}\left( x \right){{R}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+\nabla H_{i}^{\text{T}}\left( x \right){{{\bar{g}}}_{i}}\left( x \right)\omega +\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta - \\ & \nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\left( \frac{1}{2}r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{I}_{m}} \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)= \\ & -\nabla H_{i}^{\text{T}}\left( x \right){{R}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+{{\omega }^{\text{T}}}\bar{g}_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)- \\ & \frac{1}{2}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)r_{i}^{\text{T}}\left( x \right){{r}_{1}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta = \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{{\bar{g}}}_{i}}\left( x \right)\bar{g}_{i}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)- \\ & \frac{1}{2{{\gamma }^{2}}}\nabla H_{i}^{\text{T}}\left( x \right){{{\bar{g}}}_{i}}\left( x \right)\bar{g}_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+\frac{1}{2}\nabla H_{i}^{\text{T}}\left( x \right){{{\bar{g}}}_{i}}\left( x \right)\omega +\frac{1}{2}{{\omega }^{\text{T}}}\bar{g}_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)- \\ & \frac{1}{2}{{\left\| \gamma \omega \right\|}^{2}}+\frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta = \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{{\bar{g}}}_{i}}\left( x \right)\bar{g}_{i}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)- \\ & \frac{1}{2}{{\left\| \gamma \omega -\frac{1}{\gamma }\bar{g}_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right) \right\|}^{2}}+\frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta \le \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{{\bar{g}}}_{i}}\left( x \right)\bar{g}_{i}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta \\ \end{align}。 根据引理 1,可得  \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)\le -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)+\frac{1}{2}\left( \nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\nabla {{H}_{i}}\left( x \right)+{{\eta }^{\text{T}}}\eta \right)\le \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)+\frac{1}{2}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}{{\varepsilon }_{i}}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right){{\left( \frac{1}{2}r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)= \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)- \right. \\ & \left. \frac{1}{8}{{\varepsilon }_{i}}{{g}_{i}}\left( x \right){{\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+\frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)= \\ & -\nabla H_{i}^{\text{T}}\left( x \right){{T}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+\frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right) \\ \end{align}。 所以  {{\dot{H}}_{i}}\left( x \right)+\nabla H_{i}^{\text{T}}\left( x \right){{T}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right)\le \frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right)。 故对∀iΛ,γ-耗散不等式{{\dot{H}}_{i}}\left( x \right)+{{Q}_{i}}\left( x \right)\le \frac{1}{2}\left( {{\gamma }^{2}}{{\left\| \omega \right\|}^{2}}-{{\left\| z \right\|}^{2}} \right),\forall \omega 成立,其中,{{Q}_{i}}\left( x \right)=\nabla H_{i}^{\text{T}}\left( x \right){{T}_{i}}\left( x \right)\forall {{H}_{i}}\left( x \right)\ge 0,{{H}_{\infty }}控制设计目标 R1 满足。 无外部干扰(ω=0)时,闭环系统可表示为  \dot{x}=\left[ {{J}_{i}}\left( x \right)-{{R}_{i}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+{{g}_{1}}\left( x \right)\text{sat}\left( u \right)。 (9) 由式 (9) 得  {{\dot{H}}_{i}}\left( x \right)=\nabla H_{i}^{\text{T}}\left( x \right)\dot{x}=\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{J}_{i}}\left( x \right)-{{R}_{i}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\text{sat}\left( u \right)。 (10) 将式 (7) 代入式(10),并由式 (4) 得  \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)=-\nabla H_{i}^{\text{T}}\left( x \right){{R}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\eta - \\ & \nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)\left( \frac{1}{2}r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{I}_{m}} \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right) \\ \end{align}。 由引理 1可得  \begin{align} & {{{\dot{H}}}_{i}}\left( x \right)\le -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)+{{g}_{i}}\left( x \right)\left( \frac{1}{2}r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{I}_{m}} \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}\left( \nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)+{{\eta }^{\text{T}}}\eta \right)\le \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)+\frac{1}{2}{{g}_{i}}\left( x \right)\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)g_{i}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+ \\ & \frac{1}{2}{{\varepsilon }_{i}}\nabla H_{i}^{\text{T}}\left( x \right){{g}_{i}}\left( x \right){{\left( \frac{1}{2}r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right)= \\ & -\nabla H_{i}^{\text{T}}\left( x \right)\left[ {{R}_{i}}\left( x \right)+\frac{1}{2}{{g}_{i}}\left( x \right)\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)g_{i}^{\text{T}}\left( x \right)- \right. \\ & \left. \frac{1}{2}{{g}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)-\frac{1}{8}{{\varepsilon }_{i}}{{g}_{i}}\left( x \right){{\left( r_{i}^{\text{T}}\left( x \right){{r}_{i}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{i}^{\text{T}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)= \\ & -\nabla H_{i}^{\text{T}}\left( x \right){{S}_{i}}\left( x \right)\nabla {{H}_{i}}\left( x \right) \\ \end{align} 由文献[12]中的定理及其证明推导过程易知,当 ω=0 时,命题 2 也适用于系统 (6)。 在假设 1 下,若定理 2 中的条件 (2) 满足,由命题 2 可知,ω=0 时系统 (6) 在任意切换路径 λ(t) 下都是全局渐近稳定的,故 H控制设计目标 R2 满足。 4 仿真算例 给出一个仿真例子,以验证本研究对于输入饱和非线性切换 Hamilton 系统 H控制设计方法的有效性。 例 1 考虑如下输入饱和非线性切换 Hamilton 系统  \left\{ \begin{align} & \dot{x}=\left[ {{J}_{i}}\left( x \right)-{{R}_{i}}\left( x \right) \right]\nabla {{H}_{i}}\left( x \right)+{{g}_{i}}\left( x \right)\text{sat}\left( u \right)+{{{\bar{g}}}_{i}}\left( x \right)\omega , \\ & z={{r}_{i}}\left( x \right)g_{i}^{\text{T}}\left( x \right)\nabla {{H}_{i}}\left( x \right),i=1,2, \\ \end{align} \right. (11) 式中: x={{\left[ {{x}_{1}},{{x}_{2}} \right]}^{\text{T}}}\in {{\text{R}}^{2}},\text{sat}\left( u \right)\in \text{R}是控制输入,ω 是外部干扰,ri(x)=ri (i=1,2)是权矩阵,{{J}_{1}}\left( x \right)=\left( \begin{matrix} 0 & 4-x_{1}^{2}-{{x}_{2}} \\ x_{1}^{2}-4+{{x}_{2}} & 0 \\ \end{matrix} \right),\eqalign{ & {R_1}\left( x \right) = \left( {\matrix{ 0 & 0 \cr 0 & 4 \cr } } \right),{H_1}\left( x \right) = x_1^2 + {1 \over 2}x_2^2, \cr & {g_1}\left( x \right) = \left( \matrix{ 1 \hfill \cr 0 \hfill \cr} \right),{g_i}\left( x \right) = \left( \matrix{ 0 \hfill \cr 1 \hfill \cr} \right),{J_2}\left( x \right) = \left( {\matrix{ 0 & {1 - {x_2}} \cr {{x_2} - 1} & 0 \cr } } \right), \cr}${R_2}\left( x \right) = \left( {\matrix{ 2 & 0 \cr 0 & 0 \cr } } \right),{H_2}\left( x \right) = {1 \over 4}x_1^4 + x_2^4,{g_2}\left( x \right) = \left( \matrix{ 0 \hfill \cr 1 \hfill \cr} \right),{g_2}\left( x \right) = \left( \matrix{ 1 \hfill \cr 0 \hfill \cr} \right)$

 \begin{align} & {{T}_{1}}\left( x \right)={{R}_{1}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{{\bar{g}}}_{1}}\left( x \right)\bar{g}_{1}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{1}}\left( x \right)g_{1}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{1}}\left( x \right)g_{1}^{\text{T}}\left( x \right)- \\ & \frac{1}{8}{{\varepsilon }_{1}}{{g}_{1}}\left( x \right){{\left( r_{1}^{\text{T}}\left( x \right){{r}_{1}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{1}^{\text{T}}\left( x \right)=\left( \begin{matrix} \frac{1}{2{{\gamma }^{2}}}-\frac{1}{16}{{\left( \frac{1}{{{\gamma }^{2}}}+\frac{1}{4} \right)}^{2}}-\frac{1}{2} & 0 \\ 0 & 4-\frac{1}{2{{\gamma }^{2}}} \\ \end{matrix} \right)\ge 0, \\ & {{T}_{2}}\left( x \right)={{R}_{2}}\left( x \right)-\frac{1}{2{{\gamma }^{2}}}{{{\bar{g}}}_{2}}\left( x \right)\bar{g}_{2}^{\text{T}}\left( x \right)+\frac{1}{2{{\gamma }^{2}}}{{g}_{2}}\left( x \right)g_{2}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{2}}\left( x \right)g_{2}^{\text{T}}\left( x \right)- \\ & \frac{1}{8}{{\varepsilon }_{2}}{{g}_{2}}\left( x \right){{\left( r_{2}^{\text{T}}\left( x \right){{r}_{2}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{2}^{\text{T}}\left( x \right)=\left( \begin{matrix} 2-\frac{1}{2{{\gamma }^{2}}} & 0 \\ 0 & \frac{1}{2{{\gamma }^{2}}}-\frac{1}{16}{{\left( \frac{1}{{{\gamma }^{2}}}+\frac{1}{9} \right)}^{2}}-\frac{1}{2} \\ \end{matrix} \right) \\ & {{S}_{1}}\left( x \right)={{R}_{1}}\left( x \right)+\frac{1}{2}{{g}_{1}}\left( x \right)\left( r_{1}^{\text{T}}\left( x \right){{r}_{1}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)g_{1}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{1}}\left( x \right)g_{1}^{\text{T}}\left( x \right)- \\ & \frac{1}{8}{{\varepsilon }_{1}}{{g}_{1}}\left( x \right){{\left( r_{1}^{\text{T}}\left( x \right){{r}_{1}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{1}^{\text{T}}\left( x \right)=\left( \begin{matrix} \frac{1}{2{{\gamma }^{2}}}-\frac{1}{16}{{\left( \frac{1}{{{\gamma }^{2}}}+\frac{1}{4} \right)}^{2}}-\frac{3}{8} & 0 \\ 0 & 4 \\ \end{matrix} \right)>0, \\ & {{S}_{2}}\left( x \right)={{R}_{2}}\left( x \right)+\frac{1}{2}{{g}_{2}}\left( x \right)\left( r_{2}^{\text{T}}\left( x \right){{r}_{2}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)g_{2}^{\text{T}}\left( x \right)-\frac{1}{2}{{g}_{2}}\left( x \right)g_{2}^{\text{T}}\left( x \right)- \\ & \frac{1}{8}{{\varepsilon }_{2}}{{g}_{2}}\left( x \right){{\left( r_{2}^{\text{T}}\left( x \right){{r}_{2}}\left( x \right)+\frac{1}{{{\gamma }^{2}}}{{I}_{m}} \right)}^{2}}g_{2}^{\text{T}}\left( x \right)=\left( \begin{matrix} 2 & 0 \\ 0 & \frac{1}{2{{\gamma }^{2}}}-\frac{1}{16}{{\left( \frac{1}{{{\gamma }^{2}}}+\frac{1}{9} \right)}^{2}}-\frac{4}{9} \\ \end{matrix} \right)>0 \\ \end{align}

 \left\{ \begin{align} & u\left| _{\lambda \left( t \right)=1}=-2{{x}_{1}}\left( \frac{1}{2{{\gamma }^{2}}}+\frac{1}{8} \right), \right. \\ & u\left| _{\lambda \left( t \right)=1}=-4x_{2}^{3} \right.\left( \frac{1}{2{{\gamma }^{2}}}+\frac{1}{18} \right) \\ \end{align} \right.。 (12)

 \text{sat}\left( u \right)=\left\{ \begin{align} & 0.6,u>0.6, \\ & u,-0.6\le u\le 0.6, \\ & -0.6,u<-0.6 \\ \end{align} \right.。

 \lambda \left( t \right)=\left\{ \begin{align} & 2,t\in \left[ {{t}_{2k}},{{t}_{2k+1}} \right),{{t}_{2k+1}}+{{t}_{2k}}=0.15\text{s}, \\ & 1,t\in \left[ {{t}_{2k+1}},{{t}_{2k+2}} \right),{{t}_{2k+2}}-{{t}_{2k+1}}=0.15\text{s,}k=0,1,2,\cdots \\ \end{align} \right.。

 图 1 无外部干扰时系统状态 Figure 1 System states without external disturbances
 图 2 无外部干扰时系统控制器 Figure 2 System controller without external disturbances

 图 3 存在外部干扰时系统状态 Figure 3 System states with external disturbances
 图 4 存在外部干扰时系统控制器 Figure 4 System controller with external disturbances
5 结论

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